Proposition 17

Theorem

The sum of any two angles (∠ABC, ∠ACB) of a triangle (△ABC) is less than two right angles.

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Commentary

Let △ABC be a given triangle, and let ∠ABC and ∠ACB be two chosen angles.

Then ∠ABC and ∠ACB together are less than two right angles.

This proposition can be deduced from the previous proposition, Book 1 Proposition 16. A later proposition, Book 1 Proposition 32, is a stronger version of both that proposition and this one.

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Computable version

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GeometricScene[{{A.,B.,C.,O.,D.,E.,F.,G.},{}},​​{GeometricStep[{CircleThrough[{A.,B.,C.},O.],Line[{A.,O.,C.}],Line[{D.,E.}],EuclideanDistance[D.,E.]≤EuclideanDistance[A.,C.]}],​​GeometricStep[{F.∈Line[{A.,C.}],EuclideanDistance[A.,F.]EuclideanDistance[D.,E.]}],​​GeometricStep[{GeometricAssertion[{CircleThrough[{F.},A.],CircleThrough[{A.,B.,C.},O.]},{"Concurrent",G.}]}],​​GeometricStep[{Line[{A.,G.}]}]},​​{EuclideanDistance[A.,G.]EuclideanDistance[D.,E.]}​​]//RandomInstance

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Additional instances

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